Nonlinear diffusions, hypercontractivity and the optimal LP-Euclidean logarithmic Sobolev inequality

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:293

Article

Del Pino, M ; Dolbeault, J ; Gentil, I

关键词: optimal L-p-euclidean logarithmic sobolev inequality;    sobolev inequality;    nonlinear parabolic equations;    degenerate parabolic problems;    entropy;    existence;    cauchy problem;    uniqueness;    regularization;    hypercontractivity;    ultracontractivity;    large deviations;    Hamilton-Jacobi equations;   

DOI  :  10.1016/j.jmaa.2003.10.009

来源: Elsevier

【 摘 要 】

The equation u(t) = Delta(p)(u(1/(p-1))) for p > 1 is a nonlinear generalization of the heat equation which is also homogeneous, of degree 1. For large time asymptotics, its links with the optimal L-p-Euclidean logarithmic Sobolev inequality have recently been investigated. Here we focus on the existence and the uniqueness of the solutions to the Cauchy problem and on the regularization properties (hypercontractivity and ultracontractivity) of the equation using the L-p-Euclidean logarithmic Sobolev inequality. A large deviation result based on a Hamilton-Jacobi equation and also related to the L-p-Euclidean logarithmic Sobolev inequality is then stated. (C) 2003 Elsevier Inc. All rights reserved.

【 授权许可】

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